On the edge-balance index sets of B(n)

Abstract

Let G be a graph with vertex set V (G) and edge set E(G), and let Z₂ = {0, 1}. A labeling f of a graph G is said to be edge-friendly if ｜ef (0) - ef (1)｜ ≤ 1. A edge-friendly labeling f : E(G) → Z2 induces a partial vertex labeling f+ : V (G) ! Z₂ de¯ned by f+(x) = 0 if the number of edges incident on x is 0 more than 1. similarly, f+(x) = 1 if the number of edges incident on x is 1 more than 0. f+(x) is not defined if the number of edges incident on x is 1 equal to the number of edges labeled by 0. For i ∈Z₂, let vf (i) = v(i) = card{v ∈ V (G) : f+(v) = i} and ef (i) = e(i) = card{e ∈ E(G) : f(e) = i}. The edge-balance index set of the graph G, EBI(G), is defined as {｜vf (0) - vf (1)｜ :the edge labeling f is edge-friendly}. Let G be a graph and u, v be two distinct vertices of G. We construct the graph Pn(G, {u, v}) as follows. The vertex set V (Pn(G, {u, v}) is the union of n - 1 copies of V (G). We denote the vertices u, v in the copy of V (G) by ui, vi. The edge set E(Pn(G, {u, v}) is the union of n - 1 copies of E(G) with vj and uj+1 identified for j = 1, 2,.., n - 2. We call Pn(G, {u, v}) the chain sum of (G, {u, v}) by Pn. For n > 3, We denote Pn(K₄ - e, {х,z}) by B(n - 1) and Pn(K₄- e, {u, v}) by H(n - 1). In this paper, the edge-balance index set, the perfect index set and the complete index set of B(n) are de¯ned, the perfect index sets of B(n)(n = 1, 2, ..., 3) are researched.